February 2009
UILU-ENG-09-2202
DC-240
ADMISSION CONTROL AND
SCHEDULING FOR QoS GUARANTEES
FOR VARIABLE-BIT-RATE
APPLICATIONS ON WIRELESS
CHANNELS
I-Hong Hou and P. R. Kumar
Coordinated Science Laboratory
1308 West M ain Street, Urbana, 1L 61801
R E P O R T D O C U M E N TA TIO N PAG E
OMB NO. 0704-0188Form ApprovedP u b lic r e p o r tin g b u r d e n f o r th is c o lle c tio n o f in fo rm a tio n is e s tim a te d t o a v e r a g e 1 h o u r p e r r e s p o n s e , in c lu d in g th e tim e f o r re v ie w in g in s tr u c tio n s , s e a r c h in g e x is tin g d a ta s o u r c e s , g a th e r in g a n d m a in t a in in g th e d a ta n e e d e d , a n d c o m p le tin g a n d re v ie w in g th e c o lle c tio n o f in fo rm a tio n . S e n d c o m m e n t r e g a r d in g th is b u r d e n e s tim a te o r a n y o th e r a s p e c t o f th is c o lle c tio n o f in fo r m a tio n , in c lu d in g s u g g e s tio n s f o r r e d u c in g th is b u r d e n , t o W a s h in g to n H e a d q u a r te r s S e r v ic e s . D ir e c to r a te f o r in fo rm a tio n O p e r a tio n s a n d R e p o rts , 1 2 1 5 J e ffe r s o n D a v is H ig h w a y , S u ite 1 2 0 4 , A rlin g to n , V A 2 2 2 0 2 -4 3 0 2 , a n d t o th e O ffic e o f M a n a g e m e n t a n d B u d g e t, P a p e r w o r k R e d u c tio n P ro je c t ( 0 7 0 4 -0 1 8 8 ), W a s h in g to n , D C 2 0 5 0 3 . 1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED
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4. TITLE AND SUBTITLE
Admission Control and Scheduling for QoS Guarantees for Variable-Bit-Rate Applications on Wireless Channels
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I-Hong Hou and P. R. Kumar
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Coordinated Science Laboratory
University o f Illinois at Urbana-Champaign 1308 West Main Street
Urbana, Illinois 61801-2307
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UILU-ENG-09-2202 DC-240
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1 3 . ABSTRACT (Maximum 200 words)
Providing differentiated Quality of Service (QoS) over unreliable wireless channels is an important challenge for
supporting several future applications. We analyze a model that has been proposed to describe the QoS requirements by
four criteria: traffic pattern, channel reliability, delay bound, and throughput bound. We study this mathematical model and
extend it to handle variable bit rate applications. We then obtain a sharp characterization of schedulability vis-à-vis
latencies and timely throughput. Our results extend the results so that they are general enough to be applied on a wide
range of wireless applications, including MPEG Variable-Bit-Rate (VBR) video streaming, VoIP with differentiated quality,
and wireless sensor networks (WSN).
Two major issues concerning QoS over wireless are admission control and scheduling. Based on the model incorporating
the QoS criteria, we analytically derive a necessary and sufficient condition for a set of variable bit-rate clients to be
feasible. Admission control is reduced to evaluating the necessary and sufficient condition. We further analyze two
scheduling policies that have been proposed, and show that they are both optimal in the sense that they can fulfill every
set of clients that is feasible by some scheduling algorithms. The policies are easily implemented on the IEEE 802.11
standard. Simulation results under various settings support the theoretical study.
14. SUBJECT TERMS
QoS, wireless channels, admission control, scheduling
15. NUMBER OF PAGES 10
16. PRICE CODE
17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT
OF REPORT OF THIS PAGE OF ABSTRACT
U N C LA S S IFIE D U N C LAS S IFIE D U N C LAS S IFIE D U L
NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)
Prescribed by ANSI Std. 239-18 298-102
Admission Control and Scheduling for QoS Guarantees for
Variable-Bit-Rate Applications on Wireless Channels
l-Hong Hou
Department of Computer Science University of Illinois Urbana, IL, 61801, USA
[email protected]
Abstract
Providing differentiated Quality o f Service (QoS) over unreliable wireless channels is an important challenge for supporting several future applications. We analyze a model that has been proposed to de scribe the QoS requirements by four criteria: traf fic pattern, channel reliability, delay bound, and
throughput bound. We study this mathematical
model and extend it to handle variable bit rate appli cations. We then obtain a sharp characterization of schedulability vis-a-vis latencies and timely through put. Our results extend the results so that they are general enough to be applied on a wide range of wireless applications, including MPEG Variable-Bit- Rate (VBR) video streaming, VoIP with differenti ated quality, and wireless sensor networks (W SN).
Two major issues concerning QoS over wireless are admission control and scheduling. Based on the model incorporating the QoS criteria, we analytically derive a necessary and sufficient condition for a set of variable bit-rate clients to be feasible. Admission control is reduced to evaluating the necessary and sufficient condition. We further analyze two schedul ing policies that have been proposed, and show that they are both optimal in the sense that they can fulfill every set o f clients that is feasible by some schedul ing algorithms. The policies are easily implemented on the IEEE 802.11 standard. Simulation results un der various settings support the theoretical study.
1
Introduction
Digital wireless communication technology has contributed to the development o f several applica tion fields, such as Wireless Local Area Networks (W LAN) and Wireless Sensor Networks (W SN). Among other research issues, we anticipate that pro viding QoS support will be of increasing interest due to the increasing demands of delay-sensitive data traffic. Applications that require QoS support in clude video streaming, VoIP, and realtime surveil lance.
Essentially, QoS consists of providing guarantees on both delay and throughput for each flow in the system. Two important issues arise when providing QoS support. One is to determine whether the re quirements of a set of clients exceed the capacity of
P. R. Kumar
CSL and Department of ECE University of Illinois Urbana, IL 61801, USA
[email protected]
the wireless network. The other is, given that the re quirements o f the set of clients can all be satisfied by the network, to find a scheduling policy that actually does so.
In this paper, we extend a theoretical study on supporting QoS that allows us to jointly deal with fundamental difficulties. Our extension allows us to deal with the different traffic arrival patterns gen erated by different flows, which has not been ad dressed in previous work. Clients may have different demands either due to the different applications they are running, or price they are willing to pay. Even within the same flow, the generated traffic may not be periodic. Applications like MPEG video stream ing may generate traffic with variable bit rate (VBR). Thus, mechanisms based on static resource alloca tion cannot handle this kind o f traffic. We consider the different throughput requirements of the clients. Finally, a realistic theory of QoS must take into ac count the unreliable nature of wireless networks. Wireless transmissions are vulnerable to fading and shadowing effects, resulting in different qualities for each link.
We begin by providing a mathematical frame work for QoS support based on an earlier work [5]. The wireless network is described by an ab stract client-server model. The model incorporates all the aforementioned criteria: delay bounds, differ entiated throughput bounds, various traffic patterns with probabilistic packet arrival, and heterogeneous channel reliability. This abstract model can capture the realistic characteristics o f a number o f wireless applications. Based on this model, w e first derive an extension of a necessary condition for a set of clients to be feasible. We also analyze two earlier proposed dynamic scheduling policies. We analyti cally prove that both proposed policies can fulfill ev ery set of clients that satisfy the necessary condition. Thus, we not only show that the two policies are op timal but also establish that the necessary condition is indeed sufficient to flows with different arrival pat terns. Based on this finding, admission control is reduced to evaluating the necessary and sufficient condition. In summary, we jointly address both ad mission control and scheduling under the described model, extending it to handle flows with different
traffic generating patterns so that it is applicable to several scenarios of substantial interests.
In addition to the theoretical study, implementa tion issues are also discussed. We demonstrate that it is easy to implement the policies under the current IEEE 802.11 mechanism. Simulation results for both VoIP traffic and VBR video streaming are shown. The results suggest that the proposed policies are optimal in that they fulfill every feasible set of clients. They also confirm the accuracy o f the necessary and suffi cient condition.
The rest of the paper is organized as follows. Sec tion 2 summarizes some existing work on providing QoS for VBR traffic. In Section 3, we describe the ab stract client-server model with QoS criteria. Section 4 demonstrates how this model can be applied to a variety of wireless applications. In Section 5, we de rive a necessary condition for a set o f clients with dif fering traffic generation patterns to be feasible. Two proposed dynamic scheduling policies are proved to be optimal even in a more general setting in Section 7. In Section 8, we discuss how to implement the policies under the IEEE 802.11 mechanisms. Simu lation results are shown in Section 9. Finally, Section 10 concludes this paper.
2 Related Work
Providing QoS for wireless multimedia applica tions is gaining extensive research interest. Stock- hammer, Jenkac, and Kuhn [14] have studied the minimum initial delay and the minimum required buffer size for video streaming. Their study consid ers the case where there is only one wireless client in the system. Kang and Zakhor [7] have focused on improving the quality of video streaming by giv ing priorities to packets according to the content of the video. Li and Schaar [9] have proposed an adap tive algorithm for tuning the MAC retry limit for lay ered coded video. These works all lack provable per formance bounds. Wongthavarawat and Ganz [15] have studied the scheduling problem in IEEE 802.16. Their result is bound to IEEE 802.16 and not ap plicable to other MAC mechanisms. Raghunathan et al [11] and Shakkottai and Srikant [13] have derived theoretical results on minimizing the total number of expired packets in a system. Their re sults, however, cannot provide differentiated QoS for each user. Our work extends the recently proposed method o f Hou, Borkar, and Kumar [5] which deals with admission control and scheduling for the re strictive case where all clients generate traffic peri odically with the same period. Their results are not applicable to the more complicated traffic patterns. He et al [4] and Zhou et al [16] have considered providing QoS in WSNs. Their studies focus more on implementation issues rather than theoretical re sults. Fattah and Leung [3] have summarized other existing scheduling algorithms.
3 A Model For QoS With Probabilistic
Arrivals
We first describe an abstract client-server system with QoS requirements that generalizes a model that has been proposed in [5]. We w ill show that the more general model captures the characteristics o f a variety of wireless applications o f substantial inter ests, such as MPEG VBR video streaming, VoIP with differentiated quality and wireless sensor networks in Section 4.
Consider a system with N clients, numbered as {1 ,2 ,. ..,N } , and one server. Clients generate jobs for the server to accomplish. We assume that time is slotted. During each time slot, the server can at tempt exactly one job. We further assume the time slots are grouped into intervals, with each interval containing x time slots. Jobs can only be generated in the beginning o f an interval and must be finished within that interval. Unfinished jobs are discarded at the end o f an interval. Thus, a delay bound of x time slots is imposed on all jobs.
In addition to the delay bound, the QoS require ments can be further classified in three respects: traf fic pattern, reliability, and throughput. To reflect dif ferent traffic patterns for various applications, we do not restrict attention only to clients that gener ate one job during each interval as in [5]. Rather, clients generate jobs according to a probability mass
function R :
2
l 1,2.- A } (o, i). For a given subset ofclients S C {1 ,2 ,... W } , the probability that exactly every client in S generates a job in an interval is R (S). Notice that under this generic description, we do not assume clients generate jobs independently. Neither do we assume that jobs generated in an interval are independent from those in other intervals. This ex tension allows us to provide QoS to several applica tions noted above.
As in [5], we assume wireless channels are unre liable. When the server attempts to transmit a job to client n, the job gets delivered with probability
pn, which is called the reliability fo r client n . If the
attempted job is not delivered, it stays in the sys tem, and the server can further attempt it before the end o f that interval. Finally, each client n requires a long-term average throughput o f qn delivered jobs per interval. Since, on average, client n generates
Ls-.nes R{S) jobs per interval, the throughput require
ment is equivalent to a delivery ratio requirement of
Zs-nlsRis) ' That is, the fraction of discarded jobs for
client n cannot exceed 1 - ^ - •
In each time slot, the server can either choose to attempt to transmit a job in the system or to stay idle. The choice that the server takes is based on a
scheduling policy:
D
efinition 1. Let H, be the set o f all possible histories o f the system up to time slot t. A scheduling policy is a function r|: H, —► {1 ,2 ,..., N,ty} with the interpre
tation that, at time slot t + 1, the server attempts to transmit the job from client n if r\(ht) = n or idles if
r|(/i,) = <j), where h, e H, is the actual history that the
system has experienced.
The goal o f a scheduling policy is to meet the de mands o f all clients. With an unreliable system, the performance of a policy is not deterministic. Thus, a more careful specification of the requirement for performance is required, as proposed in [5]:
D
efinition2.
Aset o f clients is said to be fulfilled bya scheduling policy r\ if the long-term average through put o f each client n is at least qn jobs per interval with probability 1. That is,
K 1
lim inf Y — 1 (a job fo r client n is accomplished in the
A’^°° k= i K
kJh interval) > q„ , with probability 1 ,
fo r every client n, where 1 {■) is the indicator function.
Before obtaining such a policy that can fulfill a particular set o f clients, one needs to determine whether the set of clients is feasible:
D
efinition3.
A set o f clients is said to be feasible ifthere exists a scheduling policy r| that fulfills it.
Finally, we aim to design an optimal scheduling
policy:
D
efinition4.
An optimal scheduling policy is apolicy that fulfills every feasible set o f clients.
4 Examples o f Applications
In this section, we describe several delay-sensitive wireless applications that can be described by the ex tended model introduced in the previous section.
4.1 Video Streaming
In this scenario, each client is a wireless user, such as a laptop or a PDA, and the server is an access point (AP). Each user subscribes to a video stream and requests som quality from the AP. When pack ets, which correspond to the aforementioned jobs, for clients arrive at the AP, the AP can attempt them by transmitting a packet to the corresponding client. Upon receiving a packet, the user must reply with an ACK. Thus, the length o f a time slot is the time re quired to transmit both a data packet and an ACK. An attempt is considered successful only if the AP receives the ACK from the client. Wireless links are known to be unreliable and vary in quality from client to client. Thus, the parameter p„ captures the heterogeneous link qualities for different clients with the interpretation that whenever the server sends a packet to client n, it receives an ACK with probability
Pn-The MPEG coding algorithm is widely used to generate real-time video traffic. MPEG, along with other video coders, generates VBR traffic to main tain a fixed video quality. Depending on the context, MPEG alternates between three coding modes (I,P,B) that require different numbers of bits per frame. Given a fixed packet size, the three coding modes generate packets at different rates. Thus, the du rations between packet arrivals vary throughout the whole video. In terms o f our model, this means the
packet arrivals in each interval is a random variable, where a higher bit rate implies a higher arrival prob ability. Thus, the traffic pattern o f video streaming can be well-captured by our model.
Finally, while packet loss is inevitable, each user may require a certain delivery ratio bound, which can be converted into a throughput bound, and cap tured by the parameter qn. The different values o f
q„ for each user also reflect that the requested video
qualities vary for users.
4.2 VoIP Traffic
Like in the previous example, we have a set o f wireless users and an AP, which serves as the server. Each user requests a VoIP traffic flow from the AP. The major difference between a VoIP traffic and and a video stream is that VoIP traffic involves both up link traffic and downlink traffic. To this end, we create two clients for each user, one for uplink traf fic and one for downlink traffic. When the AP at tempts a downlink client, n, it sends a data packet for the corresponding user and waits for an ACK. The attempt is considered successful if the AP re
ceives an ACK, which happens with probability
pn.
On the other hand, when the AP attempts an uplink client, m, it first sends out a small request message to the corresponding user. The designated user, upon receiving the request message, replies with a data packet. The attempt is considered successful, with
probability
pm,
if both the request message and thedata packet are delivered. The length o f a time slot is set large enough to accommodate both an attempt for the downlink client and an attempt for the uplink client.
In this paper, we consider audio codecs that gen erate constant bit rate (CBR) traffic, such as ITU-T standards G.711 and G.718. Thus, clients generate jobs periodically, with smaller period corresponding to higher bit rate. The job generation time for clients may be offset. For example, consider a set o f three clients {1 ,2 ,3 }, where clients 1 and 2 have period 2, while client 3 has period 3. Client 1 generates jobs at intervals 1,3,..., client 2 generates jobs at inter vals 2 ,4 ,..., and client 3 generates jobs at intervals 1,4,.... Thus, we have /?({1,3}) = /?({2,3}) = 1/2. This example demonstrates how our model captures the traffic pattern of VoIP traffic.
4.3
Real Time Surveillance
We now address the scenario of a wireless sen sor network for real time surveillance. There are two levels of devices in the network: multiple sim ple sensor nodes, which corresponds to clients, and a more powerful base station, or data aggregator, that plays the role of the server. To avoid packet collisions between sensor nodes, we adopt a server centric scheme. When the server attempts a job from a particular client, it polls the corresponding sensor node for data. The attempt is successful, with prob
ability pn, if a data packet is received by the server.
One example o f such a setting is a Body Sensor Net work (BSN) [10]. In BSN, we aim to record a his
togram of physiological data for medical use. Timely packet delivery is required in cases of emergency events.
Since sensor nodes monitor a variety of events, their readings are o f differing importance. For ex ample, in the context of a BSN, there may be sen sor nodes for monitoring heart activity, blood pres sure, and body temperature. Thus, we assume each client generates jobs periodically, with the differing frequencies o f job generation reflecting the impor tance of the corresponding data. As described in the previous case, such a traffic pattern also fits into our model.
5 The Necessary Condition for Feasibil
ity
In this section, w e extend a necessary condition in [5] for a set of clients to be feasible to the more general model with variable traffic arrival patterns. Intuitively, the more often the server attempts jobs for a client, the higher the throughput that the client gets. This observation is described more formally in the following lemma:
L
emma1.
The long-term average throughput o f a client n is at least q„ jobs per interval if and only if the server, on average, attempts jobs from that client wn = times per interval.We w ill hereby refer to w„ as the implied attempt
rate fo r client n. Thus, a set o f clients is fulfilled if
and only if the average attempts per interval for jobs from each client is higher than its implied attempt rate.
Since the length of an interval is x time slots and the server can attempt jobs at most once in each time slot, the following necessary condition can be obtained:
L
emma2.
A set o f N clients is feasible only ifThis necessary condition turns out, however, to be not sufficient. Since undelivered jobs are discarded at the end of each interval, the server can only at tempt jobs that are generated in the current interval. It is possible that, at some time slot of an interval, all jobs are accomplished and the server is forced to stay idle. While the number of idle time slots depends on the scheduling policy, we show its probability distri bution is the same for a particular set o f policies.
D
efinition 5. A scheduling policy is said to be workconserving if the server never idles whenever there is
any undelivered job at the server.
L
emma3.
The probability distribution o f the amount of idle time slots in an interval is the same fo r all work conserving scheduling policies.P
roof.
Let yn be the random variable denoting thenumber of attempts the server needs to make for a job from client n before delivering it. y„ has the geometric distribution with parameter pn, that is,
Prob{yn = t} = pn{ 1 -P n Y ~ l for all positive integers t. Further, assume that a subset 5 o f clients generates
jobs in an interval. Let LsjT1 be the random variable
indicating the number of idle time slots in such an interval under scheduling policy r|. We have:
r
= f 'C-LneSYn,
if InesYn < T,{ 0, otherwise,
for all work conserving policies. Thus, the probabil ity distribution o f Ls^ is the same for all work con serving policies. □
We will hereby define Ls LsjT1, where r| is any
work conserving policy.
The following observation shows that we can al ways construct a work conserving policy, from any policy, by modifying it so that it performs at least as good as the original policy.
L
emma4.
Let r\ be a scheduling policy that fulfills some sets o f clients. Then there exists a work conserv ing policy r\' that fulfills the same set o f clients.P
roof.
The policy r\ can be modified into a workconserving one by attempting any unaccomplished job whenever r| idles. This modification cannot re duce the number of undelivered jobs for any client and thus would fulfill any set of clients that T| ful fills. □
Based on this lemma, we can therefore limit our discussion to work conserving policies throughout the rest of the paper. Suppose a subset S o f clients generates jobs in an interval, then Lemma 3 implies the expected number of idle time slots in that inter val is E[Ls]. Since such an interval occurs with prob ability R{S), the average number o f idle time slots in an interval is £ 5/?(S)£[Ls], and the server can, on
average, therefore make x - '£ sR(S)E[Ls] attempts in
an interval. This observation leads to the following refined necessary condition:
N
! > „ <
t-£R(S)£[Z.s].
CD
« = i
s
However, we can go even further by considering all subsets o f the set of all clients {1 , 2 , . . . , # }. For any subset S' C {1 ,2 ,... ,N }, let
/s.:= £ * (S )£ [max{0,x- £ y„}J
s
neSDS
1= £ * ( S )e|w |.
s
This is the average number of time slots spent
idling in an interval, if S'were the set o f all clients.
Clearly, if a set of clients is feasible, all subsets of it are also feasible. Hence, we can further refine the necessary condition (1):
L
emma5.
A set o f clients is feasible only if'Enes Wn < x —Is holds fo r every subset S.It may seem that the condition for a strict subset S o f {1 ,2 ,...,N } is redundant, and that w e only need to evaluate the condition for all clients. However, the following example shows that merely evaluating the condition (1 ) is not sufficient.
E
xample 1. Consider a system with interval length x = 3, and two clients. Each client generates one job in every interval, that is, R {{1 ,2 }) = 1. The raliabil- ities for both clients are p\ = p i = 0.5. Client 1 re quires a throughput o f q\ = 0.876, while the through put requirement o f client 2 is qi = 0.45.Now, we have:
w’i = 1.76,
h’2 = 0.9,
f { l } = / {2} = *-25,
/{i,2} = 0.25.
If we evaluate the condition for the subset o f S — {1 }, we find vt’i = 1.76 > 1.75 = x —7{i}- This indicates that the set of clients is not feasible. However, if we evaluate the condition for all clients {1 ,2 }, we have
wi + u
’2
= 2.66 < 2.75 = x — /{1)2}. Thus, this example suggests that merely evaluating the condition for all clients is not sufficient. □Surprisingly, we w ill show that the necessary con dition stated in Lemma 5 is indeed sufficient in Sec tion 7.
6 Scheduling Policies
In a previous work, Hou, Borkar, and Kumar [5] have proposed and shown that two index type o f policies are both optimal when clients generate a job in each interval. (In terms of our model, this means # ({1 ,2 ,... ,TV}) = 1, supposing there are TV clients.) Both policies are most debt first policies. In the be ginning o f each interval, the server computes the
debts owed to each client and assigns priority accord
ingly, clients with higher debts getting higher prior ity. In any time slot during this interval, the server attempts the job from the client with the highest pri ority among those who have an unaccomplished job. The only difference between the two policies is their differencing definitions on debt.
The first policy, the most time-based debt first pol
icy, attempts jobs from each client at least as often
as its implied attempt rate:
D
efinition6.
Let un( t)
denote the number o f attempts that the server makes fo r jobs from client n up to time slot t. The time-based debt fo r client n is defined to be wnt - un(t). The policy that assigns pri orities according to the time-based debts is called the
most time-based debt first policy.
The time-based debt reflects how much a client is lagging behind its implied attempt rate. By giving a client with large debt high priority, the client is, on average, granted more attempts during the interval. Thus, that client w ill have a better chance to catch up with its implied attempt rate.
The next policy adopts a more direct approach by tracking how many jobs the server actually delivers for a client:
D
efinition7.
Let cn(t) denote the number o f jobs fo rclient n accomplished by the server up to time slot t. The weighted-delivery debt fo r client n is defined to
be [qnt — cn(t )]/pn. The policy that assigns priorities according to the weighted-delivery debts is called the
most weighted-delivery debt first policy.
Note that qnt — c„(t) is the number o f jobs that the server should accomplish for client n before meet ing its demands. Moreover, since the server accom plishes a job for client n with probability pn every time it makes an attempt, the weighted-delivery debt can be thought o f as the number o f attempts that the server owes the client. Thus comes the definition o f the most weighted-delivery debt first policy.
7 Proofs o f Optimality
In this section, w e prove that these policies are also optimal for any arbitrary traffic pattern, since, as w e have noted in Section 1, many applications actually require variable bit rates.
Our proof is also based on Blackwell’s approacha- bility theorem [1],
Consider a single player game with payoff func tion M , whose value is a probability distribution in the Euclidean TV-dimensional space depending on the action taken by the player. Suppose, under some policy, the player takes action a{i) and gets a payoff v(/‘), which is an TV-dimensional vector, in each round
i. Blackwell studied the long-term average payoff the
player gets, that is, £ {=1 v (a {i))/ j, and intro
duced the concept o f approachability.
D
efinition8.
Let AC
be any set in the N -dimensional space. Consider a policy r|, which incurs payoffs v (a (l)),v (fl(2 )),.... Let 6j be the distance be tween the point £/=1 v (a (i))/ j. We shall say A is ap proachable under policy r\, if fo r every e > 0 there is a jo such that,
Prob{$j > z fo r some j > j o } < z .
In other words, the distance between the point o f the long-term average payoff and the set A converges to 0 with probability 1 .
Blackwell derived a sufficient condition for ap proachability:
T
heorem1.
Let AC
be any closed set in theTV-
dimensional space. Let r\ be a policy whose action
depends solely on the average payoff to date, xy =
l/ l/ v (a (i))/ j. Thus, we can express a { j) by
a'{xj).
Then A is approachable under rj if the following state ment holds:
If x j £ A, let y be the closest point in A to Xj and H be the hyperplane passing through y and perpendicular to the line segment xyy. A is approachable under r\ if H separates Xj and the expected payoff o f round j, that is, E [v {a '{x j))].
Based on this fundamental theorem, we prove that both most debt first policies are optimal. Since a feasible set o f clients must satisfy the necessary con dition in Lemma 5, we only need to prove that the two policies fulfill every set of clients that satisfy the necessary condition.
T
heorem2.
The most time-based debt first policy isP
roof.
We first translate the model into a single player game. A round in the game corresponds to an interval in the model. The player is the server. The action the player can take is in choosing the pri orities of clients, with the interpretation that an un accomplished job for a client is attempted only after all jobs from clients with higher priorities are accom plished. The payoff the player gets is the net change of the time-based debt owed to each client, which is thus an jV-dimensional vector. To be more precise,the payoff the player gets is
v = [vi, vj,...,
v n], wherevn equals wn minus the number o f times the server
attempts the job from client n.
By Lemma 1, the demand o f a client n is met if the server attempts its jobs at least wn = qn/pn times per interval on average, or equivalently, the
client has a non-positive time-based debt. Thus,
to establish the optimality of the most time-based debt first policy, we only need to show that the set
A := {z = [zi,z i , ■ ■ •,z n]|zn < 0, V « } is approachable un der this policy.
Suppose that at the beginning o f some interval,
the average payoff is x = [xi,X
2
,...,x v ]. If x € A,no action violates approachability by Theorem 1. If x <£ A, at least one of xi ,X
2
, ... ,x/v is strictly positive, and we can reorder the clients so that xi > X2
> • • >xm > 0 > xm+i • • • > xat. The closest point in A to x
is y = [ 0 , 0 , , 0,xm+i,x„;+2,. . . ,xn]. The hyperplane
passing through y and perpendicular to the line seg ment xv is H := ( z\h(z) := I% = l x„z„ = 0}.
Let x be the payoff of this round according to the most time-based debt first policy. Also, let wn be the number of times the sever attempts the job
from client n in the interval. We can express
x
asX = [ u 'l — VV’i , W;2 — VV’2 , . . •, Wn — vTw].
Since h(x) = 'Ln=ixn > 0, in order to show H sepa
rates
x
and£[x],
it suffices to show that h(x) < 0. Wehave: m h(x) = £ x„ (w„ - h’„) n= 1 m—1 n n = E [ ( * » - Xn+ i ) ( ! > * - £ wt] n= 1 k=l k=l ni m
+ *m (£ W*“ £vpjfc).
k=1 k= 1Next we evaluate the value of £J=1 Wk for each n. First assume that a subset S o f clients generate jobs at the beginning o f an interval. By the most time- based debt first policy, the server w ill give priority ac cording to the ordering 1 ,2 ,...,N. Hence, £J*=1 wk is the number o f attempts the server makes if there are only jobs for a subset S„ = {1 ,2 ,..., n} n 5 of clients.
In other words, Y!i=i ™k = x - LSn, where LSn is the random variable indicating the number of time slots that remain idle in an interval when only jobs from clients in the subset S„ are present. Thus w e have
E[Y!k=i Wk\S] = T - E [L s n]. Taking the expected value
over all S yields:
k= 1 k=l = '£ R (S )E ['£ w t \S] S k=l = t - £ f i ( S ) £ [ £ . s n { i,2 . ..,„ ) ] s = X - / {lf 2, ...,«}•
Now, according to the necessary condition stated
in Lemma 5, we have £J*=1h’* < x - /{
1
.2
, =Ek
= 1
Wk, for all n. Further, xi > X2
> • • • > x„, >0.
Thus, E[h{x) <
0],
and A is approachable under themost time-based debt first policy by Theorem 1, which also implies that the most time-based debt first policy is optimal. □
T
heorem3.
The most weighted-delivery debt firstpolicy is also optim al
P
roof.
Like in the previous proof, we also need totranslate this policy into one for the single player game. Again, a round in the game corresponds to an interval in our model. The action a player, which is the server, can take is to decide the priorities of clients. However, in this case, the payoff the player gets is the net change of the weighted-delivery debt. In other words, the payoff is an N-dimensional vec tor
v
= [v i,v2 , . . . ,vat] , wherev„
= (qn - 1 )/pn if the server accomplishes a job for client n in the interval, or vn — qn/pn if not. The throughput of a client n is at least qn jobs per interval if it has a non-positive weighted-delivery debt. Thus, we can prove that the most weighted-delivery debt is optimal by showing that the set A:=
{z= [zi,Z
2,---,zv]|z« < 0,Vn}
is ap proachable.Let
x
=[xi,x
2,...,xv]
be the average payoff at the beginning of an interval. Again, we only need to evaluate the performance of the most weighted-delivery debt first policy under the case
x
^ A. Wecan reorder the clients so that
xi > X
2> • • • >
xm >0
> X/n-f 1 > ■ • ■ > x n- The closest point in A tox
is y=
[0,0,...,0,xm+i,xm+2,...,xjv].
The hyperplanepassing through yand perpendicular to the line seg
ment
xy
is H := (z\h(z) :=! ”=i
xnzn =0}.
Let Kn be the indicator function that the server accomplishes a job from client n, which is a ran
dom variable. The payoff o f this interval is
x
=[(#1- K i) / Pu {Q2 - f t 2 ) / P 2 , - - - , ( q N - Xn) /p n
]-By Theorem 1, the set A is approachable if H sep
arates
x
and £[x]. Since h{x) = YHUixj, >0,
we onlyhave: h(x) = n=l ;h-1 Qn ft/i Pn n= 1 k= 1 P k k= 1 £ * 7C^-é î a- ¿ I £* 71* »i-l *=1 Pk n= 1 A=1 %k + * „ ,( L w* - L — ) ( since . . Di. = f f )rk k=l *1 Pk
Since a'i > -T
2
> • • • > Jt„, > 0, it suffices to showL'k=iwk < E [Lk=i ^ ]* for every n. Recall that the
necessary condition stated in Lemma 5 requires
S U w* < t - ^{
1
,2
, f or every n, to be feasible. Thus, w e only need to show £[ ££= 1 =
x — /{i)2...n} to establish optimality. Further, we
have ¿ ¡ j g _ , g ] = I sR ( S ) £ [ I ^ , g|S] and /{1,2... =
£sJf(S)£[i.Sn{i,
2
,-,n}]- The proof is hence completeby showing that E [Z nk=l 2t|S] = x - £ [ L sn{lf2)...,„} ] for
every 5 and n, which is done in Lemma 6 below. □
L
emma6.
Under the priority order{1,2,
E [H k = ifk\s\ = x - £ [L s n {i,
2
,...,«}]> fo r n = 1,2, and a / i5 C {l, 2 ,. .. ,iV }.P
roof.
Suppose client m doesn’t generate a job inthe interval, that is m £ S. Then the server can not make any attempt for client m, and we have £[7tm|S] = 0. Also, since m £ S, m £ ( S n { l , 2 , . . . , « } ) , and client m plays no role in deciding the value
of £[Lsn{i,
2
,...,»}]• Thus, we can delete every clientthat is not in S and reorder the remaining clients. Equivalently, w e only need to prove £[££=1 j^|5] = * - £ [ £ {
1
,2
,...,«}]> for S D {1 , 2 , . . . , « }.We prove this by induction. First consider the case
n — 1. Since client 1 has the highest priority, its job
is accomplished unless the server fails in all the x attempts. Thus,
£ [ — |S]
Pi
Prob{the job o f client 1 is accomplished} Pi
1 —(1 —£ i ) T
Pi
On the other hand, we also have: T— 1
£ [£ {i}] = Prob{the job from client 1 is
accomplis-t=i
hed in at most x - 1 attempts}
= £ ( • - ( ! - P i ) ' - ' )
: H - , r
Pi
This gives us £[^j-|S] = x - £ [ £ { i j ] and the lemma holds for the case n — 1.
Assume that £ [ E Li jfk\s\ = T ~ £ [£ {
1
,2
, hol dsfor all n < m. Consider the case n — m + 1. Since the client m + 1 has the lowest priority among clients {1 ,2 ,...,/ «+ 1}, its job is attempted only after all jobs from client 1 through client m are accomplished. Since there are £ {
1
,2
,...,»i} time slots left after the server accomplishes jobs from the first m clients, we have:£[n»»+i ^ ¿ {i,2 , ....»,}
-= P ro b {the job o f client m + 1 is accomplished in
x - a attempts} = 1 - ( 1 - £ » « +i
)x~ct-On the other hand, since £ {
1
,2
,....»,} - £ { i ,2
,...,m+i} is the number o f attempts that the server makes for a job from client m + 1, we also have:£[¿{
1,
2,...,»,} ~ ¿ {
1,
2,...,»
1+
1} |£{i,
2,...,»,} = °]
X
= J2 Prob{the server makes at least t - o attempts
/=o+l
for the job from client m +
1
}= i (i - p. +i) '- '’ - 1
t=o+ 1
1 “ ( ! - £ » , + l]
= £|îî±i|S,i(1A
=
0],
£»1+1 Pm+l
for all a. Thus, \S\ = £ [ £ {1)2,...,m} - £ {1,2,...,»,+!}]•
Finally, we have:
= £ [ £ ^ | S ] + £ [ ^ ± i | S ]
A=1 Pk Pm+l
=X —£ [£ {lt2,...,m}] +£[£{1,2,...,»,} ~£{1,2,...,»i+1}] —X -£ [£ {1 )2,...,»,+
1}]-By induction, the lemma holds for all n. □ A final remark is that, since both policies fulfill every set of clients that satisfy the necessary condi tion in Lemma 5, this condition is also sufficient for feasibility.
T
heorem4.
A set o f clients is feasible if and only if 'L„es wn < t — Is holds fo r every subset S.8 Implementation on IEEE 802.11
While our generic model can be applied to a va riety of wireless applications, we believe that the WLANs can particularly benefit most from our study due to their wide deployment, and increasing de mands of QoS support. In this section, we show that the proposed policies can be easily implemented in the current IEEE 802.11 mechanisms.
The IEEE 802.11 defines two transmission modes for Medium Access Control (MAC) [6]: the Dis tributed Coordination Function (DCF) and the Point Coordination Function (PCF). The two mechanisms can coexist by dividing a superframe, which corre sponds to the interval in our model, into a PCF contention-free period (CFP) followed by a DCF con tention period (CP). The PCF mechanism is server centric, and thus suitable for implementing the pro posed policies.
In the PCF mode, when the server schedules a downlink transmission, it sends out the data to a client after sensing the channel being idle for a pe riod of PIFS. The client, after receiving the packet, waits a period of SIFS and then replies with an ACK. When the server schedules an uplink transmission, it sends out a CF-POLL packet, which contains in formation about which client is scheduled to trans mit, after the channel is idle for a period o f PIFS. The designated client replies with a data packet, or a NULL packet if it does not have any data to send, after waiting for a period o f SIFS upon receiving the CF-POLL packet. On the other hands, any node that operates in the DCF mode must wait for the channel being idle for at least a period of DIFS before trans mitting a packet. The value of DIFS is set larger than both the values of SIFS and PIFS. Thus, the server is granted the highest priority when it operates in the PCF mode.
9 Simulation Results
We have implemented both most debt first poli cies, namely, the most time-based debt first policy and the most weighted-delivery debt first policy, on ns-2 under the IEEE 802.11 PCF mechanism. We evaluate the performance o f these two policies un der two scenarios, one for the MPEG video stream ing traffic and one for the VoIP traffic. We com pare the most debt first policies with the naive ap proach of using the IEEE 802.11 DCF standard and a
random priority policy that also operates in the PCF
mode. The random priority policy works like the most debt policies, only that the server assigns prior ities to clients randomly at the beginning o f each in terval. We define a metric, throughput insufficiency, to reflect the difference of the desired throughput and the actual throughput. To be more specific, let
d„(t) be the actual average throughput of client n at
some time slot t. The throughput insufficiency o f client n is qn — dn(t) if qn > dn(t) or 0 otherwise. The
throughput insufficiency o f the system is the sum of the throughput insufficiency over all clients.
9.1 VoIP Traffic
We follow the ITU-T G.729.1 [12] codec, which generate traffic with bit rates ranging from 8 kbits/s
to 32 kbits/s, in simulating VoIP traffic. We as
sume an interval length o f 20 ms and 160 Bytes VoIP packet. IEEE 802.11b is used as the underlying MAC protocol. Related parameters are described in Table 1. Under this setting, the transmission times for both the uplink traffic, consisting of a CF-POLL packet and a data packet, and the downlink traffic, consisting of a data packet and an ACK, are slightly less than 610
ps, allowing 32 time slots in an interval.
Table 1: Simulation Setup For VoIP
Interval 20 ms
Payload size per packet 160 Bytes
Transmission data rate 11 Mb/s
SIFS 10 ps
PIFS 30 ps
DIFS 40 ps
We consider two groups o f clients, group A and group B. Each client in group A generates packets periodically with period 60 ms, resulting in a 21.3 kbits/s flow, and requires 99% delivery ratio. Clients in group B also generate packets periodically but with period 40 ms, which corresponds to 32 kbits/s flows, and require 80% delivery ratio. The starting times o f clients in each group are separated evenly. To be more specific, we can further divide the two
groups into subgroups A 1} A2, A3, B\, and Bi. Clients
in subgroup Aj generate packets at the beginning of
intervals /, i + 3, i +
6,
..., while clients in subgroupBj generate packets at the beginning of intervals j, j + 2, j + 4 ,.... Evaluating the necessary and suf
ficient condition in Theorem 4 suggests that a set of 6 clients in each of the subgroup A, and 5 clients in each of Bj is feasible while a set o f 6 clients in each of Aj and 6 clients in each of Bj is not.
Figure la shows the simulation results for the aforementioned feasible set o f clients on the four tested policies, namely, the two most debt first poli cies, the random policy, and the DCF mechanism. The throughput insufficiencies of both most debt first policies converge to 0 over time, showing that they fulfill this set of clients. However, the most weighted-delivery debt first policy converges much faster than the most time-based debt first policy. This is because the weighted-delivery debt reflects die ac tual throughput a client is having, and thus is a more direct and precise measure than the time-based debt. While the most time-based debt first policy may be easier to implement, the most weighted-delivery debt first policy should be preferred when tight per formance is important. The other two policies both fail to fulfill this set o f clients, indicating that they
(a) Performance of a feasible set
assume the system uses the higher data rate IEEE 802.11a. Some related parameters are shown in Ta ble 2. Under this setting, the transmission time for a data packet and an ACK is roughly 650 jus, allowing 9 time slots in an interval.
Table 2: Simulation Setup For Video Streaming
Interval 6 ms
Payload size per packet 1500 Bytes
Transmission data rate 54 Mb/s
SIFS 16 ¡is
PIFS 25 /us
DIFS 34 fis
Tim e (sec)
(b) Performance of an infeasible set
Figure 1: Throughput insufficiency for VoIP traffic
cannot be optimal. The random policy, though also operating in the contention-free PCF mode, fails to fulfill the set o f clients because it does not consider the differentiated requirements of the clients. The DCF mechanism has the worst performance among the four since it suffers greatly from contentions and collisions.
To verify the correctness of the necessary and suf ficient condition in Theorem 4, w e also run simula tion on the predicted infeasible set o f clients com posed o f 6 clients in each subgroup Aj and 6 clients in each subgroup Bj. The results are shown in Fig ure lb. All the four tested policies of course fail to fulfill this set o f clients, since it is indeed infea sible. Further, it can be noted that although the two most debt first policies do not fulfill this set of clients, they still incur less throughput insufficiency than the other two policies. This result shows that the pro posed policies perform w ell in comparison to some other policies even when dealing with an infeasible set o f clients.
9.2 MPEG Video Streaming
We consider the case where a number of wireless users request MPEG video stream traffic with vari ous requirements from the AP. Since video stream re quires much larger bandwidth than VoIP traffic, we
Some previous works [8] [2] model the MPEG VBR traffic by a Markov chain consisting of three ac tivity states. Traffic with different bit rates is gen erated in each o f the three states. Martin et al [2] hence derived statistical results for the movie “The
Graduate” . We adopt their model for the MPEG
VBR traffic and transfer the statistical result into the packet arrival probability in each interval under our setting, which is summarized in Table 3.
Table 3: MPEG Traffic Pattern
Activity Great T l i i T Regular
Arrival probability 1 0.8 0.75
We assume there are two groups of clients, A and
B. Clients in group A require high quality video that
generates packet according to Table 3 and a 90% de livery ratio for each packet, resulting in a requested throughput o f 0.765 packet per interval. Clients in group B only require low quality video that generates packets only 80% as often as those in group A and demand a 60% delivery ratio, or a throughput re quirement o f 0.408 packet per interval. The channel reliability of the n,h client in each group is (60 + n)%. By evaluating the necessary and sufficient condition in Theorem 4, we predict that a set of 4 group A clients and 4 group B clients is feasible, while a set of 5 group A clients and 4 group B clients is not.
Figure 2a shows the simulation results on the fea sible set o f clients composed by 4 group A clients and 4 group B clients. Like in the case of VoIP traffic, the throughput insufficiency o f both the two most debt policies converge to zero over time, and the two poli cies therefore fulfill this set of clients. Also, the most weighted-delivery debt first policy converges faster than the most time-based debt first policy. The ran dom policy and DCF, on the other hand, fail to fulfill this set o f clients. In the case o f video streaming, the AP is the only wireless device that generates traffic. Thus, there should be no contention and collision in the system. Still, the CSMA/CA mechanism used by the DCF forces the AP to backoff a random time before each transmission. This overhead results in
1.2 ! o. z 06 3 a. “ 0.4 £ X 0.2 - W e ig h te d -d e live ry - — Time-based - — - Random — * — DCF H t H A A A A A A A A A A A A A A Tim e (se c)
(a) Performance of a feasible set
1.6 $1.4 1 1.2 W e ig h te d -d e live ry j — — Tim e -b ase d - — - Random —
*
— DCF1
1 .3 0.8
Q
.
X
? o.e
2P 0-4
0.2
0
0 1 2 3 4 5 6 Tim e (sec)(b) Performance of an infeasible set
Figure 2: Throughput insufficiency for video stream ing
much higher throughput insufficiency for DCF, than that for the random policy.
Simulations on the infeasible set consisting of 5 group A clients and 4 group B clients are also con ducted, and the results are shown in Figure 2b. All the four tested policies of course fail to fulfill this set of clients. These results also demonstrate that our model can be applied to a wide range o f applications. Finally, the two most debt first policies have the least throughput insufficiency among the four tested poli cies, showing that they offer good performance even for an infeasible set o f clients.
10 Conclusions
We have analytically addressed the problem of providing QoS support for heterogeneous VBR traf fic flows over the unreliable wireless channels. We study an extension of a proposed mathematical model that incorporates delay bounds, throughput bounds, traffic patterns, and channel reliabilities. This extended model turns out to adequately cap ture the characteristics o f a variety of wireless appli cations, including video streaming, VoIP, and BSN. Based on the model, we have derived a necessary and sufficient condition for a set of clients to be fea sible. Admission control is thus reduced to evalu
ate the necessary and sufficient condition. We have also studied the scheduling problem, and studied two scheduling policies. We prove that these two policies are both optimal in the sense that they fulfill every feasible set of clients. In addition to theoret ical study, we have also addressed implementation issues under IEEE 802.11 and implemented the two scheduling policies in ns-2. Simulation results have confirmed our theoretical studies and shown that the proposed scheduling policies outperform other tested policies under a variety o f settings.
11
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